# §10.60 Sums

## §10.60(i) Addition Theorems

Define $u$, $v$, $w$, and $\alpha$ as in §10.23(ii). Then with $\mathop{P_{n}\/}\nolimits$ again denoting the Legendre polynomial of degree $n$,

 10.60.1 $\frac{\mathop{\cos\/}\nolimits w}{w}=-\sum_{n=0}^{\infty}(2n+1)\mathop{\mathsf% {j}_{n}\/}\nolimits\!\left(v\right)\mathop{\mathsf{y}_{n}\/}\nolimits\!\left(u% \right)\mathop{P_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\alpha\right),$ $|ve^{\pm i\alpha}|<|u|$.
 10.60.2 $\frac{\mathop{\sin\/}\nolimits w}{w}=\sum_{n=0}^{\infty}(2n+1)\mathop{\mathsf{% j}_{n}\/}\nolimits\!\left(v\right)\mathop{\mathsf{j}_{n}\/}\nolimits\!\left(u% \right)\mathop{P_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\alpha\right).$
 10.60.3 $\frac{e^{-w}}{w}=\frac{2}{\pi}\sum_{n=0}^{\infty}(2n+1)\mathop{{\mathsf{i}^{(1% )}_{n}}\/}\nolimits\!\left(v\right)\mathop{\mathsf{k}_{n}\/}\nolimits\!\left(u% \right)\mathop{P_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\alpha\right),$ $|ve^{\pm i\alpha}|<|u|$.

## §10.60(ii) Duplication Formulas

 10.60.4 $\mathop{\mathsf{j}_{n}\/}\nolimits\!\left(2z\right)=-n!z^{n+1}\sum_{k=0}^{n}% \frac{2n-2k+1}{k!(2n-k+1)!}\mathop{\mathsf{j}_{n-k}\/}\nolimits\!\left(z\right% )\mathop{\mathsf{y}_{n-k}\/}\nolimits\!\left(z\right),$
 10.60.5 $\mathop{\mathsf{y}_{n}\/}\nolimits\!\left(2z\right)=n!z^{n+1}\sum_{k=0}^{n}% \frac{n-k+\frac{1}{2}}{k!(2n-k+1)!}{\left({\mathop{\mathsf{j}_{n-k}\/}% \nolimits^{2}}\!\left(z\right)-{\mathop{\mathsf{y}_{n-k}\/}\nolimits^{2}}\!% \left(z\right)\right)},$
 10.60.6 $\mathop{\mathsf{k}_{n}\/}\nolimits\!\left(2z\right)=\frac{1}{\pi}n!z^{n+1}\sum% _{k=0}^{n}(-1)^{k}\frac{2n-2k+1}{k!(2n-k+1)!}{\mathop{\mathsf{k}_{n-k}\/}% \nolimits^{2}}\!\left(z\right).$

## §10.60(iii) Other Series

 10.60.7 $\displaystyle e^{iz\mathop{\cos\/}\nolimits\alpha}$ $\displaystyle=\sum_{n=0}^{\infty}(2n+1)i^{n}\mathop{\mathsf{j}_{n}\/}\nolimits% \!\left(z\right)\mathop{P_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits% \alpha\right),$ 10.60.8 $\displaystyle e^{z\mathop{\cos\/}\nolimits\alpha}$ $\displaystyle=\sum_{n=0}^{\infty}(2n+1)\mathop{{\mathsf{i}^{(1)}_{n}}\/}% \nolimits\!\left(z\right)\mathop{P_{n}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\alpha\right),$
 10.60.9 $e^{-z\mathop{\cos\/}\nolimits\alpha}=\sum_{n=0}^{\infty}(-1)^{n}(2n+1)\mathop{% {\mathsf{i}^{(1)}_{n}}\/}\nolimits\!\left(z\right)\mathop{P_{n}\/}\nolimits\!% \left(\mathop{\cos\/}\nolimits\alpha\right).$
 10.60.10 $\mathop{J_{0}\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\alpha\right)=\sum_{% n=0}^{\infty}(4n+1)\frac{(2n)!}{2^{2n}(n!)^{2}}\mathop{\mathsf{j}_{2n}\/}% \nolimits\!\left(z\right)\mathop{P_{2n}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\alpha\right).$
 10.60.11 $\sum_{n=0}^{\infty}{\mathop{\mathsf{j}_{n}\/}\nolimits^{2}}\!\left(z\right)=% \frac{\mathop{\mathrm{Si}\/}\nolimits\!\left(2z\right)}{2z}.$ Symbols: $\mathop{\mathrm{Si}\/}\nolimits\!\left(\NVar{z}\right)$: sine integral, $\mathop{\mathsf{j}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z}\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable A&S Ref: 10.1.52 Referenced by: §10.60(iii), Ch.10 Permalink: http://dlmf.nist.gov/10.60.E11 Encodings: TeX, pMML, png See also: Annotations for 10.60(iii)

For $\mathop{\mathrm{Si}\/}\nolimits$ see §6.2(ii).

 10.60.12 $\displaystyle\sum_{n=0}^{\infty}(2n+1){\mathop{\mathsf{j}_{n}\/}\nolimits^{2}}% \!\left(z\right)$ $\displaystyle=1,$ Symbols: $\mathop{\mathsf{j}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z}\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable A&S Ref: 10.1.50 Referenced by: §10.60(iii) Permalink: http://dlmf.nist.gov/10.60.E12 Encodings: TeX, pMML, png See also: Annotations for 10.60(iii) 10.60.13 $\displaystyle\sum_{n=0}^{\infty}(-1)^{n}(2n+1){\mathop{\mathsf{j}_{n}\/}% \nolimits^{2}}\!\left(z\right)$ $\displaystyle=\frac{\mathop{\sin\/}\nolimits\!\left(2z\right)}{2z},$ Symbols: $\mathop{\sin\/}\nolimits\NVar{z}$: sine function, $\mathop{\mathsf{j}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z}\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable A&S Ref: 10.1.51 Referenced by: §10.60(iii) Permalink: http://dlmf.nist.gov/10.60.E13 Encodings: TeX, pMML, png See also: Annotations for 10.60(iii) 10.60.14 $\displaystyle\sum_{n=0}^{\infty}(2n+1)(\mathop{\mathsf{j}_{n}\/}\nolimits'\!% \left(z\right))^{2}$ $\displaystyle=\tfrac{1}{3}.$ Symbols: $\mathop{\mathsf{j}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z}\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable Referenced by: §10.60(iii) Permalink: http://dlmf.nist.gov/10.60.E14 Encodings: TeX, pMML, png See also: Annotations for 10.60(iii)

For further sums of series of spherical Bessel functions, or modified spherical Bessel functions, see §6.10(ii), Luke (1969b, pp. 55–58), Vavreck and Thompson (1984), Harris (2000), and Rottbrand (2000).

## §10.60(iv) Compendia

For collections of sums of series relevant to spherical Bessel functions or Bessel functions of half odd integer order see Erdélyi et al. (1953b, pp. 43–45 and 98–105), Gradshteyn and Ryzhik (2000, §§8.51, 8.53), Hansen (1975), Magnus et al. (1966, pp. 106–108 and 123–138), and Prudnikov et al. (1986b, pp. 635–637 and 651–700). See also Watson (1944, Chapters 11 and 16).